重力子が質量をもつ重力理論と二つの計量をもつ重...

重力子が質量をもつ重力理論と二つの計量をもつ重力理論

Shin’ichi Nojiri(Шинъичи　Нозири)

Department of Physics &

Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI),

Nagoya Univ.

– Typeset by FoilTEX – 1

Preliminary

“bigravity’’ = system of massive spin 2 field (massive graviton)+ gravity (includes massless spin 2 field = graviton)

Fierz-Pauli action (linearized or free theory), 3/4 century agoM. Fierz and W. Pauli, “On relativistic wave equations for particles of arbitrary spin in an

electromagnetic field,” Proc. Roy. Soc. Lond. A 173 (1939) 211.

The Lagrangian of the massless spin-two field (graviton) hµν is given by

L0 = −1

2∂λhµν∂

λhµν+∂λhλµ∂νh

µν−∂µhµν∂νh+

1

2∂λh∂

λh ,(h ≡ hµ

µ

).

Massless graviton: 2 degrees of freedom (helicity),Massive graviton: 5 degrees of freedom (2s+ 1, spin s = 2).


The Lagrangian of the massive graviton with mass m is given by

Lm = L0 −m2

2

(hµνh

µν − h2)

(Fierz-Pauli action) .

When m = 0, gauge symmetry (linearized general covariance)

δhµν = ∂µξν + ∂νξµ ,

ξµ(x): space-time dependent gauge parameter.

The combination hµνhµν − h2:

Fierz-Pauli tuning (not related with any symmetry)

For the combination hµνhµν − (1 − a)h2,

if a = 0, there appears ghost scalar field with mass

m2g =

3 − 4a

2am

2(m

2g → ∞when a → 0

)


Hamiltonian and counting of degrees of freedom:D(D−1)

2 − 1 propagating degrees of freedom in D dimensions(5 degrees of freedom for D = 4).

Legendre transformation only with respect to the spatial components hij.

πij =∂L∂hij

= hij − hkkδij − 2∂(ihj)0 + 2∂kh0kδij ,

⇒ S =

∫dDx

{πijhij −H+ 2h0i (∂jπij) +m2h2

0i

+h00

(∇2hii − ∂i∂jhij −m2hii

)},

H =1

2π2ij −

1

2

1

D − 2π2ii +

1

2∂khij∂khij − ∂ihjk∂jhik

+ ∂ihij∂jhkk −1

2∂ihjj∂ihkk +

1

2m2(hijhij − h2

ii) .


m = 0 case: h0i, h00: Lagrange multipliers → constraints

∂jπij = 0 , ∇2hii − ∂i∂jhij = 0 .

First class constraints → gauge symmetry (⇐ general covariance)

For D = 4, hij and πij each have 6 components, respectively.→ 12 dimensional phase space.

4 constraints + 4 gauge invariances→ 4 dimensional phase space

(two polarizations (helicities) of massless graviton)


m = 0: h0i are no longer Lagrange multipliers δh0i ⇒ h0i = − 1m2∂jπij,

S =

∫dDx{πijhij − H + h00

(∇2

hii − ∂i∂jhij − m2hii

)},

H =1

2π

2ij −

1

2

1

D − 2π

2ii +

1

2∂khij∂khij − ∂ihjk∂jhik

+ ∂ihij∂jhkk −1

2∂ihjj∂ihkk +

1

2m

2(hijhij − h

2ii

)+

1

m2(∂jπij)

2.

h00: Lagrange multiplier → single constraint

C = −∇2hii + ∂i∂jhij + m

2hii = 0 ,

Secondary constraint:

{H, C}PB =1

D − 2m

2πii + ∂i∂jπij = 0 , H =

∫ddx H ,


Two second class constraints.

For D = 4,12 dimensional phase space − 2 constraints = 10 degrees of freedom(5 polarizations of the massive graviton and their conjugate momenta).


vDVZ(van Dam, Veltman, and Zakharov) discontinuity

H. van Dam and M. J. G. Veltman, “Massive and massless Yang-Mills and gravitational fields,” Nucl. Phys.

B 22 (1970) 397.

V. I. Zakharov, “Linearized gravitation theory and the graviton mass,” JETP Lett. 12 (1970) 312 [Pisma

Zh. Eksp. Teor. Fiz. 12 (1970) 447].

Discontinuity of m → 0 limit in the free massive gravity with the Einsteingravity due to the extra degrees of freedom in the limit.⇒ the Vainstein mechanismA. I. Vainshtein, “To the problem of nonvanishing gravitation mass,” Phys. Lett. B 39 (1972) 393.

Non-linearity screens the extra degrees of freedom (non-linearity becomesstrong when m is small).


Boulware-Deser ghost

D. G. Boulware and S. Deser, “Classical General Relativity Derived from Quantum Gravity,” Annals Phys.

89 (1975) 193.

In non-linear (interacting) theory, 6th degree of freedom appears as a ghost.

Non-linear massive gravity action with flat metric ηµν, hµν = gµν − ηµν

S =1

2κ2

∫dDx

[√−gR− 1

4m2ηµαηνβ (hµνhαβ − hµαhνβ)

].

t = t0 + dt

t = t0

x: constant

6-

Ndt

Nidt ADM formalism (N : lapse function, Ni: shift function)

g00 = −N2+ g

ijNiNj , g0i = Ni , gij = gij .

i, j, · · · = 1, 2, 3, gij: inverse of the spatial metric gij.


m = 0 case

Einstein-Hilbert action (after partial integrations)

1

2κ2

∫dDx√gN[(d)

R − K2+ K

ijKij

],

(d)R: curvature of spatial metric gij, Kij: extrinsic curvature

Kij =1

2N(gij − ∇iNj − ∇jNi) ,

∇i: covariant derivative w.r.t. the spatial metric gij.Canonical momenta with respect to gij:

pij

=δL

δgij=

1

2κ2

√g(K

ij − Kgij)

,


Hamiltonian:

H =

(∫Σt

ddx pabgab

)− L =

∫Σt

ddx NC +NiCi .

C =√g[(d)R+K2 −KijKij

], Ci = 2

√g∇j

(Kij −Khij

),

Kij =2κ2

√g

(pij −

1

D − 2phij

).

For m = 0, Hamiltonian vanishes. N , Ni: Lagrange multipliers⇒ C = 0, Ci = 0: first class constraints ⇔ general covarianceIn D = 4,12 phase space metric components − 4 constraints − 4 gauge symmetries

= 4 phase space degrees of freedom= degrees of freedom in linearized theory of massless spin 2 graviton


m = 0 case (hij ≡ gij − δij)

ηµαηµβ (hµνhαβ − hµαhµβ)

= δikδjl (hijhkl − hikhjl) + 2δijhij − 2N2δijhij + 2Ni

(gij − δij

)Ni ,

Action

S =1

2κ2

∫dDx

{pabgab −NC −NiCi

− m2

4

[δikδjl (hijhkl − hikhjl) + 2δijhij

−2N2δijhij + 2Ni

(gij − δij

)Nj

]}.

N2, NiNj terms ⇒ N2, Ni: Not Lagrange multipliers but auxiliary fields.


N =C

m2δijhij, Ni =

1

m2

(gij − δij

)−1 Cj .

No constraints nor gauge symmetries.Hamiltonian:

H =1

2κ2

∫ddx

{1

2m2

C2

δijhij+

1

2m2Ci(gij − δij

)−1 Cj

+m2

4

[δikδjl (hijhkl − hikhjl) + 2δijhij

]}.

12 phase space degrees of freedom, or 6 real degrees of freedom.One more degree of freedom, compared with linearized theory

⇒ ghost scalar

Boulware-Deser ghost


Massive gravity without ghost

C. de Rham and G. Gabadadze, “Generalization of the Fierz-Pauli Action,” Phys. Rev. D 82, 044020 (2010)

[arXiv:1007.0443 [hep-th]],

C. de Rham, G. Gabadadze and A. J. Tolley, “Resummation of Massive Gravity,” Phys. Rev. Lett. 106

(2011) 231101 [arXiv:1011.1232 [hep-th]].

S. F. Hassan and R. A. Rosen, “Resolving the Ghost Problem in non-Linear Massive Gravity,” Phys. Rev.

Lett. 108 (2012) 041101 [arXiv:1106.3344 [hep-th]].

Non-dynamical metric fµν (∼ ηµν),√g−1f :

√g−1f

√g−1f = gµλfλν

Minimal extension of Fierz-Pauli action:

S = M2p

∫d4x

√−g

[R− 2m2 (tr

√g−1f − 3)

].


⇒ vDVZ discontinuity ⇒

S = M2p

∫d4x

√−g

[R+ 2m2

3∑n=0

βn en(√

g−1f)

],

e0(X) = 1 , e1(X) = [X] , e2(X) = 12([X]

2 − [X2]) ,

e3(X) = 16([X]

3 − 3[X][X2] + 2[X3]) ,

e4(X) = 124([X]

4 − 6[X]2[X2] + 3[X2]2 + 8[X][X3]− 6[X4]) ,

ek(X) = 0 for k > 4 ,

X = (Xµν) , [X] ≡ Xµ

µ ,

∼ Galileon ⇒ Vainshtein mechanism(longitudinal scalar mode (hµν ∼ ∂µ∂νϕ) ∼ Galileon scalar field)


Hamiltonian constraint: Minimal extension caseADM formulation, fµν = ηµν ⇒

L =πij∂tγij +NR0 +N iRi − 2m2√γ N(tr√

g−1η − 3).

(g−1η)µν =1

N2

(1 N lδlj

−N i (N2γil −N iN l)δlj

), N i = γijNj .

Highly nonlinear action in Nµ ⇒ New combinations ni

N i = (δij +NDij)n

j ,

Dij : (

√1− nT In)D =

√(γ−1 −DnnTDT )I ,

I = δij , I−1 = δij ,


⇒ L =πij∂tγij +NR0 +Ri(δij +NDi

j)nj

− 2m2√γ[√

1− nT In+Ntr (√γ−1I−DnnTDT I)− 3N

].

Linear in N .δni ⇒ ni = −Rjδ

ji[4m4 det γ +Rkδ

klRl

]−1/2: Not including N .

δN ⇒ R0 +RiDijn

j − 2m2√γ[√

1− nrδrsns Dkk − 3

]= 0 .

+ secondary constraint = 2 constraints.

12 components of γij and πij − 2 constraints= 10 components (massive spin 2)


Bimetric gravity (bigravity)

S. F. Hassan and R. A. Rosen, “Bimetric Gravity from Ghost-free Massive Gravity,” JHEP 1202 (2012) 126

[arXiv:1109.3515 [hep-th]].

Dynamical fµν (background independent).

S =M2g

∫d4x√− det g R

(g)+M2

f

∫d4x√

−det f R(f)

+ 2m2M2eff

∫d4x√

−det g

4∑n=0

βn en(√g−1f) ,

1/M2eff ≡ 1/M2

g + 1/M2f .

R(g): scalar curvature for gµν, R

(f): scalar curvature for fµν.


Spectrum of the linearized theory

Minimal case: β0 = 3, β1 = −1, β2 = 0, β3 = 0, β4 = 1.

Linearize gµν = gµν +1

Mghµν , fµν = gµν +

1

Mflµν ,

⇒ S =

∫d4x (hµνEµναβhαβ + lµνEµναβlαβ)

− m2M2eff

4

∫d4x

[(hµ

ν

Mg− lµν

Mf

)2

−(hµ

µ

Mg−

lµµMf

)2].

Eµναβ: usual Einstein-Hilbert kinetic operator.


Change of variables

1

Meffuµν =

1

Mfhµν +

1

Mglµν ,

1

Meffvµν =

1

Mghµν −

1

Mflµν .

⇒

S =

∫d4x (uµνEµναβuαβ + vµνEµναβvαβ)

− m2

4

∫d4x

(vµνvµν − vµµv

νν

).

One massless spin-2 particle uµν and one massive spin-2 particle vµν withmass m.


Renormalizable theory of massive spin two particleY. Ohara, S. Akagi and S. Nojiri, “Renormalizable theory of massive spin two particle and new bigravity,”

arXiv:1402.5737 [hep-th].

New ghost free interactionsK. Hinterbichler, “Ghost-Free Derivative Interactions for a Massive Graviton,” JHEP 1310 (2013) 102

[arXiv:1305.7227 [hep-th]].

Ld,n ∼ηµ1ν1···µnνn∂µ1

∂ν1hµ2ν2

· · · ∂µd−1∂νd−1

hµdνdhµd+1νd+1

· · ·hµn+d/2nun+d/2,

hµν =gµν − ηµν ,

“pseudo linear terms

ηµ1ν1µ2ν2 ≡η

µ1ν1ηµ2ν2 − η

µ1ν2ηµ2ν1 ,

ηµ1ν1µ2ν2µ3ν3 ≡η

µ1ν1ηµ2ν2η

µ3ν3 − ηµ1ν1η

µ2ν3ηµ3ν2 + η

µ1ν2ηµ2ν3η

µ3ν1

− ηµ1ν2η

µ2ν1ηµ3ν3 + η

µ1ν3ηµ2ν1η

µ3ν2 − ηµ1ν3η

µ2ν2ηµ3ν1 .


Linear with respect to h00 in the Hamiltonian.

Do not appear the terms which include both of h00 and h0i.

Variation of h00 ⇒ a constraint for hij and their conjugate momenta πij

⇒ eliminate the ghost.

Power-counting renormalizable model of the massive spin two particle

Lh0 =1

2ηµ1ν1µ2ν2µ3ν3

(∂µ1

∂ν1hµ2ν2

)hµ3ν3

−m2

2ηµ1ν1µ2ν2hµ1ν1

hµ2ν2

−µ

3!ηµ1ν1µ2ν2µ3ν3hµ1ν1

hµ2ν2hµ3ν3

−λ

4!ηµ1ν1µ2ν2µ3ν3µ4ν4hµ1ν1

hµ2ν2hµ3ν3

hµ4ν4

=1

2(h□h − h

µν□hµν − h∂µ∂νhµν − hµν∂

µ∂νh + 2h

ρν ∂

µ∂νhµρ)

−m2

2

(h2 − hµνh

µν)−

µ

3!

(h3 − 3hhµνh

µν+ 2h

νµ h

ρν h

µρ

)−

λ

4!

(h4 − 6h

2hµνh

µν+ 8hh

νµ h

ρν h

µρ − 6h

νµ h

ρν h

σρ h

µσ + 3 (hµνh

µν)2)

.


m, µ: parameters with the dimension of massλ: dimensionless parameters.⇒ power-counting renormalizable (free from ghost)

Propagator

Dmαβ,ρσ =

1

2 (p2 +m2)

{PmαρP

mβσ + Pm

ασPmβρ −

2

D − 1PmαβP

mρσ

},

Pmµν ≡ηµν +

pµpνm2

.

p2 → ∞ ⇒ Dmαβ,ρσ ∼ O

(p2)· · · Not be renormalizable


Similar problem in the model of massive vector field

L = −1

4(∂µAν − ∂νAµ) (∂

µA

ν − ∂νA

µ) −

1

2m

2A

µAµ .

Propagator

Dµν = −1

p2 + m2P

mµν ,

which is the inverse of

Oµν ≡ −

(p2+ m

2)ηµν

+ pµpν, O

µνDνρ = δ

µρ .

p2 → ∞ ⇒ Dµν ∼ O(1): Not renormalizable

If the vector field, however, couples only with the conserved current Jµ (∂µJµ = 0)

⇒ pµpν

m2 in Pmµν drops

⇒ Dµν ∼ O(1/p2

)⇒ maybe renormalizable.


Instead of imposing the conservation law, we may add the term 2αϕ∂µAµ

Inverse of the operator

OAϕ =

(Oµν −iαpµ

iαpν 0

),

is given by

DAϕ =

(− 1

p2+m2Pνρ −i pναp2

ipραp2

m2

α2p2

), Pµν ≡ ηµν − pµpν

p2,(

OAϕDAϕ =

(δµρ 00 1

)).

Pµν = Pmµν on shell, p2 = −m2

p2 → ∞ ⇒ Propagator between twoAµ’s O(1/p2

)⇒ Renormalizable if the interaction terms are also renormalizable.


Adding the term 4αAµ∂νhµν(Oµν,αβ −iα (pµηαν + pνηαµ)

iα(pαηµβ + pβηµα

)0

)(Dαβ,ρσ −iEσ,αβiEα,ρσ Fασ

)

=

(12

(δµρδ

νσ + δµαδ

νβ

)0

0 δµσ

).

⇒

Dαβ,ρσ = −1

2 (p2 + m2)

{PαρPβσ + PασPβρ −

2

D − 2PαβPρσ

},

Eα,ρσ =1

2αp2

{pρPασ + pσPαρ −

m2pα

(D − 2) (p2 + m2)Pρσ +

pαpρpσ

p2

},

Fασ =m2

2α2p2Pασ +

(D − 1)m4

4α2 (D − 2) (p2)2(p2 + m2)

pαpσ .

Propagator between two hµν’s O(1/p2

)⇒ maybe renormalizable.


New bigravity

Couples with gravity ∼ new bigravity

S =

∫d4x

√−g

{1

2gµ1ν1µ2ν2µ3ν3∇µ1∇ν1hµ2ν2hµ3ν3 −

1

2m2gµ1ν1µ2ν2hµ1ν1hµ2ν2

−µ

3!gµ1ν1µ2ν2µ3ν3hµ1ν1hµ2ν2hµ3ν3 −

λ

4!gµ1ν1µ2ν2µ3ν3µ4ν4hµ1ν1hµ2ν2hµ3ν3hµ4ν4

+4αAµ∇νhµν} ,

hµν is not the perturbation in gµν but hµν is a field independent of gµν.Cosmology with the Einstein-Hilbert action:

SEH =1

2κ2

∫d4x

√−gR .


Assume hµν = Cgµν, C: constant

S = −∫

d4x√−gV (C) , V (C) ≡ 6m2C+4µC3+λC4 , (∇ρgµν = 0) .

C ⇐ V ′(C) = 0.Parametrize m2 and µ by

m2 =λ

3C1C2 , µ = −λ

3(C1 + C2) .

⇒ C = 0, C1, C2

V (C1) =λ

3C3

1 (−C1 + 2C2) , V (C2) =λ

3C3

2 (−C2 + 2C1) .

V (C) ∼ cosmological constant.


Summary

• Brief review on massive gravity and bigravity.

• Proposition of a renormalizable theory describing massive spin twoparticle.

• The coupling of the theory with gravity → a new kind of bimetric gravityor bigravity.

– The field of the massive spin two particle plays the role of thecosmological constant.


F (R) bigravity


重力子が質量をもつ重力理論と 二つの計量をもつ重...

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重力子が質量をもつ重力理論と二つの計量をもつ重...